How can we expand the operator $$O_{12} = \frac{1}{r^3}[({\bf \sigma_{1}\cdot{\bf r}})({\bf \sigma_{2}\cdot{\bf r}}) - r^2{\bf \sigma_{1}}\cdot {\bf \sigma_{2}}]$$ where ${\bf \sigma} = (\sigma_{x}, \sigma_{y}, \sigma_{z})$ and ${\bf r}$ is an ordinary 3-vector. Subscript 1, 2 refers to the first and second spin.
I have tried using the relation $(\sigma . {\bf a})(\sigma . {\bf b})$ but that only involves one of the spins.
Can someone help me?
Without context, it's hard to see what you are up to, but here is a standard identity you should be able to derive/confirm, $$ (({\bf \sigma_{1} + \sigma_{2})\cdot{\bf r}})^2 = ({\bf \sigma_{1}\cdot{\bf r}})^2+({\bf \sigma_{2}\cdot{\bf r}})^2 +2({\bf \sigma_{1}\cdot{\bf r}})({\bf \sigma_{2}\cdot{\bf r}}) \leadsto \\ ({\bf \sigma_{1}\cdot{\bf r}})({\bf \sigma_{2}\cdot{\bf r}})=\tfrac {1}{2} ( ({\bf \sigma_{1} + \sigma_{2})\cdot{\bf r}} )^2- 2r^2. $$
The sigma matrices are the generators of the SU(2) Lie group. There is a general relationships for such generators given by the structure constants. If my memory does not fail me, it reads $$ [\tau^a,\tau^b]= if^{abc}\tau^c . $$ In the case of the SU(2) group the structure constant is just the totally antisymmetric tensor $\epsilon^{abc}$.
